1. 3020 Differentials and Linear Approximation
AB Calculus

2. Related Rates :
How fast is y changing as x is changing?-
Differentials:
How much does y change as x changes?

3. Approximation A. Differentials
Goal: Answer Two Questions - How much has y changed? and What is y ‘s new value?

4. IF…
My waist size is 36 inches
IF I increases my radius 1 inch, how much larger would my belt need to be ?
2𝜋 𝑖𝑛𝑐ℎ𝑒𝑠
The earths circumference is 24, miles.
IF I increases the earth’s radius 1 inch, how much larger would the circumference be ?
2𝜋 𝑖𝑛𝑐ℎ𝑒𝑠

5. The Change in Value : The Differential
The Differential finds a QUANTITY OF CHANGE !
REM:
∆𝑥=4 3
NOTE: Finds the change in y NOT the value of y

6. The Change in Value : The Differential
The Differential finds a QUANTITY OF CHANGE !
REM:
∆𝑥=3
9 4
NOTE: Finds the change in y NOT the value of y

7. The Change in Value : The Differential
The Differential finds a QUANTITY OF CHANGE !
REM:
∆𝑥=2
6 4
NOTE: Finds the change in y NOT the value of y

8. The Change in Value : The Differential
The Differential finds a QUANTITY OF CHANGE !
REM:
∆𝑥= 3 2
9 8
NOTE: Finds the change in y NOT the value of y

9. Differentials and Linear Approximation in the News

10. Algebra to Calculus dy: The DIFFERENTIAL: “How much has y changed?”
“the first difference in y for a fixed change in x ”
Notation:
dy:
Also written: df

11. The Differential finds a QUANTITY OF CHANGE !
In Calculus dy approximates the change in y using
the TANGENT LINE. 1
3 4 3
3 4 3
NOTE: APPROXIMATES the change in y
The smaller the ∆𝑥 the better the approximation

12. The Differential Function: Example 1
Find the Differential Function and use it to approximate change.
A). Find the differential function.
B). Approximate the change in y
at with
𝑑𝑦= cos 𝑥 𝑑𝑥
𝑑𝑦 𝑑𝑥 = cos 𝑥
𝑑𝑦=cos⁡( 𝜋 6 ) 𝜋 36
𝑑𝑦= ∗ 𝜋 36 = 𝜋
≈.0756

13. The Differential Function: Example 2
Find the Differential Function and use it to approximate the volume of latex in a spherical balloon with inside radius
and thickness
A). Find the differential function.
B). Approximate the change in V.
C). Find the actual Volume.
𝑑𝑉 𝑑𝑟 =4𝜋 𝑟 2
𝑑𝑉=4𝜋 𝑟 2 (𝑑𝑟)
𝑑𝑉=4𝜋 =4𝜋≈12.566
4 3 𝜋 𝑟 3 = 4 3 𝜋 − 4 3 𝜋 4 3
− =12.764

14. B: Linearization
“Make It Linear!”

15. L(x) = f(a) + f / (a) (x – a)
Linearization:
Linearization:
y – y1 = m (x – x1 )
y = y1 + m (x – x1 )
L(x) = f(a) + f / (a) (x – a)
The standard linear approximation of f at a
The point x = a is the center of the approximation

16. Linearization 𝑦= 3 2 𝑦=sin⁡(𝑥) 𝑦 ′ = 1 2 𝑦 ′ =cos⁡(𝑥) L 𝑥 =𝑦+𝑦′(𝑥−𝑎)𝑦=
𝑦=sin⁡(𝑥)
𝑦 ′ = 1 2
𝑦 ′ =cos⁡(𝑥)
L 𝑥 =𝑦+𝑦′(𝑥−𝑎)
L 𝑥 = 𝑥− 𝜋 3 𝑥−

17. Linearization 𝑦= 2𝑥−1 𝑦 5 =3 𝑦 ′ = 1 2 2𝑥−1 −1 2 ∗2 𝑦 ′ = 1 2𝑥−1
𝑦= 2𝑥−1
𝑦 5 =3
𝑦 ′ = 𝑥−1 −1 2 ∗2
𝑦 ′ = 1 2𝑥−1
𝑦 ′ 5 = 1 3
𝐿 𝑥 = 𝑥−5

18. C: Tangent Line Approximation
#1 find friendly #
#2 ∆𝑥
What is the new value?
#3 y
𝜋 6 = 6𝜋 36
given
friendly
#4 y’
y2 – y1 = m ( x2 – x1 )
y2 = y1 + m (Δx)
𝜋 4 = 9𝜋 36
#5 values
𝜋 3 = 12𝜋 36

19. Linear Approximation - Tangent Line Approximation
EXAMPLE: .
Find friendly #
Wants the VALUE!
𝑎= 12𝜋 36 𝑜𝑟 𝜋 3
𝑓 12𝜋 −𝜋 36 ≈ cos 𝑥 +(−sin⁡(𝑥)( −𝜋 36 )
∆𝑥 𝑔𝑖𝑣𝑒𝑛 −𝑓𝑟𝑖𝑒𝑛𝑑𝑙𝑦
𝑦 𝜋 3 = 1 2
≈ − −𝜋 36
𝑦=cos⁡(𝑥)
≈ 𝜋 72
𝑦 ′ =−sin⁡(𝑥)
𝑦 ′ 𝜋 3 = − 3 2
𝑐𝑜𝑠 11𝜋 36 ≈ 𝜋 72
∆𝑥=− 𝜋 36

20. Linear Approximation - Tangent Line Approximation
EXAMPLE: Approximate .
Wants the VALUE!

21. II. Error

22. ERROR: There are TWO types of error: A. Error in measurement tools
- quantity of error
- relative error
- percent error
B. Error in approximation formulas
- over or under approximation
- Error Bound - formula

23. A. Error in Measurement Tools

24. Choose either or 2
.00,000,1
.00,000,2

25. EXAMPLE 1: Measurement (A)
Volume and Surface Area: The measurement of the edge of a cube is found to be 12 inches, with a possible error of 0.03 inches. Use differentials to approximate the maximum possible error in computing:
the volume of a cube
the surface area of a cube
find the range of possible measurements in parts (a) and (b).

26. EXAMPLE 2: Measurement (A)
Volume and Surface Area: the radius of a sphere is claimed to be inches, with a possible error of .02 inch
Use differentials to approximate the maximum possible error in calculating the volume of the sphere.
Use differentials to approximate the maximum possible error in calculating the surface area.
Determine the relative error and percent error in each of the above.

27. EXAMPLE 3: Measurement (B) : Tolerance
Area: The measurement of a side of a square is found to be
15 centimeters.
Estimate the maximum allowable percentage error in measuring
the side if the error in computing the area cannot exceed 2.5%.

28. EXAMPLE 4: Measurement (B) : Tolerance
Circumference The measurement of the circumference of a circle is found to be 56 centimeters.
Estimate the maximum allowable percentage error in measuring
the circumference if the error in computing the area cannot
exceed 3%.

29. B. Error in Approximation Formulas

30. ERROR: Approximation Formulas
Error = (actual value – approximation)  either Pos. or Neg.
Error Bound = | actual – approximation |
For Linear Approximation:
The Error Bound formula is
Since the approximation uses the TANGENT LINE
the over or under approximation is determined by the
CONCAVITY (2ndDerivative Test)

31. In Calculus dy approximates the change in y using the
TANGENT LINE.
The ERROR depends on distance from center( ) and the bend in the curve ( f ” (x))

32. Example 5: Approximation
Error = (actual value – approximation)  either Pos. or Neg.
Error Bound = | actual – approximation |
For Linear Approximation:
The Error Bound formula is
EX: Find the Error in the linear approximation of

33. Example 6: Approximation
EX: Find the Error in the linear approximation of

34. Last Update:
11/04/11

35. New Value : Tangent line Approximation
In words: _____________________________________________
With the differential :

36. Algebra to Calculus
How much has y changed?

37. Algebra to Calculus
How much has y changed?